MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfun Structured version   Visualization version   GIF version

Theorem zfun 6903
Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfun 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfun
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-un 6902 . 2 𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥)
2 elequ2 2001 . . . . . . 7 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
3 elequ1 1994 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3anbi12d 746 . . . . . 6 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
54cbvexv 2274 . . . . 5 (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧))
65imbi1i 339 . . . 4 ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
76albii 1744 . . 3 (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
87exbii 1771 . 2 (∃𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 220 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  uniex2  6905  axunndlem1  9361
  Copyright terms: Public domain W3C validator